Labor-Managed vs Profit-Maximizing Monopsony in the Labor Market D. S. Olgin University of Belgrade and Center for Liberal-Democratic Studies, Belgrade Corresponding address. Djordje Suvakovic Olgin, East-West Institute, Stevana Sremca 4, 11000 Belgrade, Serbia and Montenegro, Tel/Fax +381 11 322 2036, Email: [email protected], [email protected] , [email protected]The first email address is much more reliable than the others; still, please use all three addresses, and send three separate identical messages, to confirm the arrival of my packet. I should also be most grateful to you if you send this email confirmation as soon as possible, say by Monday, June 23 rd - for the right reasons, the grantors have approved the 6 months, no cost, extension of my grant, which expires by the end of June. I will of course notify you as soon as I re-establish a full control of my email addresses, but I cannot commit myself to finish this job before that critical June, 23 rd . ACKNOWLEDGMENTS. This research was supported by a grant from CERGE-EI Foundation under a program of the Global Development Network. Additional funds for grantees in the Balkan countries have been provided by the Austrian Government through WIIW, Vienna. All opinions expressed are those of the author and have not been endorsed by CERGE-EI, WIIW, or the GDN. I thank Milan Drazic and Pavle Petrovic for helpful comments and suggestions. The usual disclaimer applies.
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Labor-Managed vs Profit-Maximizing Monopsony in the Labor Market
D. S. Olgin
University of Belgrade and Center for Liberal-Democratic Studies, Belgrade Corresponding address. Djordje Suvakovic Olgin, East-West Institute, Stevana Sremca 4, 11000 Belgrade, Serbia and Montenegro, Tel/Fax +381 11 322 2036, Email: [email protected], [email protected] , [email protected] The first email address is much more reliable than the others; still, please use all three addresses, and send three separate identical messages, to confirm the arrival of my packet. I should also be most grateful to you if you send this email confirmation as soon as possible, say by Monday, June 23rd - for the right reasons, the grantors have approved the 6 months, no cost, extension of my grant, which expires by the end of June. I will of course notify you as soon as I re-establish a full control of my email addresses, but I cannot commit myself to finish this job before that critical June, 23rd. ACKNOWLEDGMENTS. This research was supported by a grant from CERGE-EI Foundation under a program of the Global Development Network. Additional funds for grantees in the Balkan countries have been provided by the Austrian Government through WIIW, Vienna. All opinions expressed are those of the author and have not been endorsed by CERGE-EI, WIIW, or the GDN. I thank Milan Drazic and Pavle Petrovic for helpful comments and suggestions. The usual disclaimer applies.
Labor-Managed vs Profit-Maximizing Monopsony in the Labor Market
ABSTRACT
This paper compares the efficiency of a labor-managed and a profit-maximizing firm (LMF
and PMF) at the monopsonistic labor market.
We demonstrate that, both locally and globally, a LMF can (efficiency) dominate a
PMF, where the local and global dominance are respectively linked with a single inverse labor
supply or wage function and a single family of such functions.
While the global dominance concept is novel, and has no counterpart in the literature,
the result on the local dominance is, to a certain extent, complementary to the well-known
possibility result on a LMF’s welfare superiority to a PMF in monopolistic competition
(Neary 1985, 1992). The issue of how to privatize non-wage-taking firms is addressed.
JEL Classification Numbers: J42, J54, L20
Keywords. Labor Market; Labor-Managed vs Profit-Maximizing Monopsony;
Efficiency Dominance.
1
Labor-Managed vs Profit-Maximizing Monopsony in the Labor Market
I. Introduction
The labor-controlled or, more familiar (though less precise), labor-managed (LM) firms are
normally linked with various organizational and property rights structures, provided that
control rights are vested in a firm’s labor.1
Initially, labor-managed firms (LMFs) used to be identified with the Western type
producer cooperatives and partnerships in the service sector (Meade, 1972; Bonin (1984);
Dreze, 1989), collective farms of the Soviet Union (Domar, 1966), and almost all industrial
firms of the ‘self-managed’ era in the SFR Yugoslavia (Ward, 1958; Vanek, 1970; Estrin,
1983).
Nowadays, LMFs may also be linked with many employee-controlled firms that have
emerged during the transition process in Russia, Ukraine, Latvia, Georgia, Belarus and
Slovenia and, to a smaller extent, in Poland, Estonia, Hungary, Romania and Bulgaria
(Lissovolik, 1997; Uvalic, 1997; Uvalic and Vaughan-Whitehead, 1997; Jones et al., 1998;
Earle and Estrin, 1996).
Finally, some forms of partial labor control and/or risk bearing, like codetermination,
internal bargaining or wage cuts in the ailing firms2, are becoming a fact in the not negligible
number of (previously) conventional proprietorships of industrialized economies.
Starting with Ward (1958), it is most often assumed that one of the basic features of
LMFs is their maximand of income per labor unit or of a ‘full’ wage (see, for example, Bonin
and Putterman, 1987; Estrin, 1983; Ireland and Law, 1982; Jossa and Cuomo, 1997), which
clearly distinguishes such enterprises from conventional, profit-maximizing firms (PMFs) 3.
1 The excellent, comprehensive survey of the vast literature on LMFs is given in Bonin and Putterman (1987). See also the review monograph by Bartlett and Uvalic (1986) and a more recent book by Jossa and Cuomo (1997). For a concise review, focusing on a certain gap between the theory and evidence on LMFs, see Bonin, Jones and Putterman (1993). A fully-fledged textbook on the LM firms is Ireland and Law (1982). 2 The first allocation suboptimal model of a codetermined firm has been constructed by Svejnar (1982). For some extensions of the model, performed within the internal-bargaining framework, see Miyazaki (1986). 3 In this connection it is worth mentioning that the so far most systematic empirical study of the LMF behavior (Pencavel and Craig, 1994) has at least not rejected the wage-maximizing hypothesis; but see also Bonin, Jones and Putterman (1993) and Craig and Pencavel (1993). Some more recent examples of adopting the wage-maximizing assumption are Baniak (2000), Futagami and Okamura (1996), and Neary and Ulph (1997).
2
As regards the comparison of LMFs’ and PMFs’ performance under various
conditions, it seems to be the dominant view that the wage-maximizing behavior by labor-
managed firms is, at least on average, inferior to traditional profit-maximization.
Still, a significant number of the results have been obtained - mostly, if not
exclusively, within the price taking environments - where the LMF behavior and its effects
are, fully or partially, equalized with those of conventional profit-maximizing firms.4
However, few cases have also been detected (Neary, 1985; Neary, 1992; Neary and
Ulph, 1997)5 which show that - under some forms of the output market imperfections - LMFs
can, in one way or another, be superior to PMFs.
The aim of this paper is to point to an additional example of a LMF’s possible
superiority to a PMF, which refers to an important case of the input market imperfection, that
of the monopsony in the labor market or, more broadly, to the non-wage-taking firms, which
face an upward sloping wage curve6.
Thus, in a sense, the present paper is due to Domar’s (1966) model of, effectively, a
labor-managed (LM) monopsony in the labor market. However, while Domar was interested
in the comparative statics of such an enterprise, our sole concern is its efficiency, as compared
with that of a corresponding profit-maximizing (PM) monopsonistic firm.
The structure of the paper is as follows.
In part II we first define, in section 1, a typical family of increasing and convex
inverse labor supply or wage functions - obtained by systematically varying the degree of
labor scarcity - which enable both a LMF and a PMF to earn nonnegative profit.
To motivate the reader not interested in the labor-management per se, we then
introduce one numerically generated graphic of this family to focus, in section 2 of part II, on
the (always existing) family-member wage function that yields exactly the Pareto optimal
equilibrium of a no loss making LMF. Of course, this surely means that in the considered case
a LM monopsony Pareto dominates a PM monopsony, since the latter, as is well-understood,
can never reach the Paretian norm.
In part III we represent the well-known PM monopsony equilibrium in the form
appropriate for straightforward efficiency comparisons with the corresponding LM
4 See, for instance, Bonin (1981;1984), Domar (1966), Dreze (1989), Estrin (1982), Greenberg (1979), McCain (1977), Miyazaki and Neary (1983). 5 See also Futagami and Okamura (1996). 6 A valuable initial source for assessing the relevance of the non-wage-taking phenomenon is Boal and Ransom (1997).
3
equilibrium. Then, we formally characterize the two types of the latter equilibrium, first
considered by Domar (1966).
In part IV we show that the family of wage functions, defined in part II, is always
divided, by some neutral member-function, in its upper and lower subfamily, where the
former implies the efficiency dominance of a LMF over a PMF, while within the latter the
converse is true.
In part V we discuss some of the results of numerical simulations, performed to obtain
an idea about the relative size of the LMF and the PMF dominance regions, and fully
presented in Appendix 2. In Appendix 1, we graphically represent the three numerical
simulations analyzed in part V, which test the sensitivity of the LMF/PMF dominance relation
on the curvature of the considered wage functions.
Summary and conclusion, where the latter also addresses the issue of privatising a
non-wage-taking enterprise, are left for part VI.
II. The Typical Family of Wage Functions and the Case When a LM Monopsony Pareto Dominates a PM Monopsony 1. The S family of inverse labor supply or wage functions
In order to define the one-parameter family of all inverse labor supply or wage functions,
which yields nonnegative profit to a non-wage-taking firm, we first introduce the function of
firm’s (non-capital) income per unit of labor or a ‘full’ wage, y:
LCLXy −
=)( (1)
where X(L) and L are the short-run production function and the labor input, and where, by
suitably choosing the measure of X, its (constant) price, p, is normalized to unity. Finally, C
stands for fixed (capital) costs.
The reader familiar with the theory of a labor-managed firm (LMF) - see, for example,
Dreze (1989), Bonin and Putterman (1987), Ireland and Law (1982) - will recognize in (1) the
most frequently assumed objective function of such an enterprise. Here, the y function -
depicted in figures 1 and 2 below – will, inter alia, serve to define the steepest wage function
that yields zero profits both to a labor-managed and to a conventional, profit-maximizing firm
(PMF).
4
In the monopsonistic labor market the typical (inverse) labor supply or wage function
faced by a firm may be represented as:
Wk = f(L, ak) ≡ Wk(L) , L > 0 , (2)
where Wk is a wage rate or a supply price of labor, L is a firm’s demand for labor and ak is
(nonnegative) parameter, which represents a measure of labor scarcity experienced by a firm.
In what follows ak will be varied so as to cover all relevant degrees of labor scarcity,
displayed in relation (3) and the related part of the text.
The f function is further characterized as follows:
0>′≡∂∂ fLf (2a)
02
2≥′′≡
∂
∂ fL
f (2b)
0>≡∂∂
kak
faf (2c)
0>′≡∂
′∂ka
kf
af (2d)
In (2a) the positivity of f’ says that the wage rate is increasing in the demand for
labor.
In (2b) the nonegativity of f’’ says that – perhaps, due to a rising marginal disutility of
labor - the wage function is convex or, at least, linear.
In (2c) the positivity of means that the wage rate is increasing with labor scarcity,
for any given demand for labor.
kaf
Finally, in (2d) the positivity of kaf ′ means that an increase in the labor scarcity
makes a greater increase in the wage rate, given any (infinitesimal) increase in the demand for
labor.
By varying the ak parameter within the interval defined in (3) below, we obtain the
one-parameter family of Wk functions, denoted by S,
5
S = {Wk = f(L, ak) W≡ k(L) , ak ∈ (ae, az)} , (3)
where the S family is bounded from below by the horizontal entry-wage schedule We,
depicted in figure 17,
We = f(L, ae) = const > 0 , (4)
and where ae generates the equilibrium labor use by a hypothetical, wage taking PMF, Le,
Thus, as already mentioned, relation (4) defines the hypothetical case of a wage-taking
enterprise - i.e., of zero labor scarcity faced by a single firm - while (5) defines the steepest
relevant wage function which, by definition, yields zero profit both to a PMF and a LMF – see
the Wz(L) function of figure 1.
7 Note that in figure 1 – as in most of simulations performed in part V - we assume, for simplicity, that the entry-wage is insensitive to the value of the ak parameter. This however does not affect our main result, on the alternating dominance of a LMF and a PMF, summarized by proposition 1 of part IV, nor does it influence the possibility result on a LMF’s (PMF’s) global dominance over a PMF (LMF), obtained via numerical simulations fully displayed in Appendix 2, and partially reported and discussed in part V below.
6
0 3
2
X’ y
WmWz
We
Ly
Wn
LmLz L
E
Mym
Mm
ΠLm
Z
We
e
N
X’0
PLm=
Figure 1. The S family of wage functions of (3), represented by the shaded area bordered by the horizontal line We of (4) and the Wz function of (5). The functions Wn and Wm appear in (17) and (12), while labor’s marginal cost, Mm, is: Mm = Wm + LW’m. The functions y and X’ are those of (1) and (7). The M point is defined by the unconstrained maximum of y, ,
given in (14a), and the corresponding (maximal) value of y, y = y
ymL
m: ; the PMF
labor use, L
),(M mym yL=
e, is of (4a). The Pareto optimal labor use, LP, identical in the considered case with the LM monopsony labor use, - see also section 2 of this part - is defined by eq. (5b), and for the typical family-member function, W
(quadratic wage functions W(L), entry-wage We = 0.4)
The complete results - which, for We = 0.4, also include linear and cubic wage
functions - are presented in table 1 of Appendix 2.
11 The values of sijδ of relations (34) – (39) and of Appendix 2 are approximate, where the computational error can be made arbitrarily small. The complete calculation procedure is available from the author on request.
20
Finally, we represent the LMF/PMF dominance relation results - analogous to that of
(34) - obtained for the entry-wage We = 0.63, and the entry-wage higher than the average
The second simulation of (38), which yields δ = δ22 = 8.10, is graphically presented in
figure 3.
22
0 3
1
Figure 3. The LMF and the PMF dominance regions, identified with the Sy and SП subfamilies of S, are approximated by the darker and lighter shaded areas, bordered by the Wz, Wn, and We functions: The case of 2nd simulation of (38), δ = δ22 = 8.10. X = 2L – 0.2L2 - production function X’ = 2 – 0.4L - labor’s marginal product y = 2 – 0.2L – C/L - income per worker C = 0.68 - fixed costs We = 0.63 - entry wage horizontal line, ak = 0 ak - the (varying) labor scarcity parameter, ),0( zk aa ∈
Wk = We + akL2 - typical wage function Wn = We + anL2 - neutral wage function that implies equal equilibrium of LMF and PMF, an = 0.0624 Wz = We + azL2 - zero-profit wage function that implies zero-profit and equal equilibrium of LMF and PMF, az = 0.564
Similar to the case of Type 1 simulations, few notes are in order.
First, as in the case of Type 1 simulations, it appears that a LMF strongly dominates
PMF.
23
Second, we observe that the change in the LMF/PMF dominance ratio is again a
systematic one – in the sense that this ratio increases by switching from families typical of
convex wage functions with smaller curvature, to families with more convex wage functions.
Third, greater LMF/PMF dominance ratios are associated with smaller values of the
entry-wage12.
VI. Summary and Conclusion
In this paper we have used a standard model of the monopsony in the labor market to compare
the efficiency of a labor-managed and conventional, profit-maximizing firm (LMF and PMF)
in non-wage-taking environments.
To accomplish this, we have first defined the local efficiency dominance, according to
which one firm dominates the other when, for a single inverse labor supply or wage function,
the former produces more output than the latter, provided that both firms are able to make no
losses.
For a well-behaved, increasing and convex typical wage function, we have then
systematically varied a suitably defined labor scarcity parameter from zero to its zero-profit
level. Given a turned U-shaped income-per-worker schedule, the latter level defines the
steepest wage curve that yields zero profit both to a LMF and a PMF, and thus have the
tangency point with the above schedule.
This procedure has generated the continuous family of wage functions, which all
ensure nonnegative profit to a LMF and a PMF and where, by definition, the number of such
functions is infinite.
Finally, we have demonstrated that this family is always divided by, some neutral
member-function, in its upper and lower subfamily, where for any function of the former a
LMF (locally) dominates a PMF, while for any function of the latter the converse is true.
Thus, we have also shown that, on the level of a single wage function, a LMF can efficiency
dominate a PMF, and vice versa.
After detecting this alternating LMF/PMF dominance relation, we have focused on
getting the idea about the relative size of the LMF and the PMF dominance regions, 12 To test the relevance of the performed simulations, we have also done the three modified exercises, where the (previously parametric) entry-wage has been modeled as an increasing function of the labor scarcity parameter. These new simulations have been designed so as to be fully comparable with the three arguably most relevant parametric entry-wage simulations, summarized by relation (38). However, it has emerged that these additional exercises have not altered the tenor of the previous results - the LMF dominance region has, on average, decreased pretty modestly, from 89% to 87%.
24
identified with the ratio of shares of the corresponding subfamilies in the above defined
family of wage functions.
To achieve this, we have had to temporarily assume that this family is discrete, and
that its member-functions (the number of which is big) are evenly spread across the family.
Also, this has required to establishing the concept of global dominance, where one firm has
been defined to globally dominate the other when the former locally dominates the latter for
more than a half of all wage functions which constitute the (entire) family.
After that, we have performed 27 (carefully selected) numerical simulations, which
combine three types of technology, three types of wage functions, and three levels of the
entry-wage.
First, the simulations indicate that the LMF/PMF dominance relation - identified with
the ratio of the LMF and the PMF dominance region - systematically increases by switching
from technologies with concave labor’s marginal product to those characterized by convex
labor’s marginal productivity. Second, the LMF dominance region also clearly (relatively)
increases by switching from families of linear wage functions, to families of (strictly) convex
functions with smaller curvatures and, finally, to families that consist of more convex (wage)
functions. Third, the above ratio is greater for lower levels of the entry wage.
The basic result of the performed simulations is that, on average, a LMF (strongly)
globally dominates a PMF, where the average size of the LMF dominance region amounts to
94% of all considered wage functions, and where just one of 27 simulations yields a
(relatively weak) PMF’s dominance - see relation (A2.2) and tables 1-3 of Appendix 2.
Finally, two notes are in order.
The first one refers to the (novel) concept of the global efficiency dominance, which
should obviously not be restricted to the present LMF/PMF case of monopsonistic labor
markets and could, in principle, be applied in various situations and under different market
structures.
However, in the present case, and when considered on the empirical level, the concept
would require each family-member function to be weighted by the probability of its
occurrence at the specific labor market. Still, on the theoretical level the (implicitly) assumed
equal probability of all relevant wage functions is acceptable, if not for the fact that all these
functions enable both a LMF and a PMF to make no losses and thus, almost by definition,
should be non-discriminatory taken into account when comparing the (global) efficiency of a
labor-managed and a profit maximizing monopsony.
25
The second note may be of relevance for the theory and policy of privatizing non-
wage-taking firms. If, say, in the context of post-socialist transition, the econometric evidence
reveals the local dominance of some insider-controlled firm (assumed to behave like a
canonical LMF) over the corresponding outsider-privatized PMF, a higher local efficiency of
the former - due to its objective of wage maximization - ought to be weighed against the
possibly superior technical productivity of the latter, observed, for example, in the case of the
outsider-privatized firms across Central-Europe.13 This, among other things, should be taken
into account when defining the strategy of how to privatize a non-wage-taking firm.
In any case, and irrespective of these remarks, the key result of the paper clearly points
to the fact that in non-wage-taking environments, and with equal technical and market
opportunities, the labor-managed firm can be more efficient, both locally and globally, than
the conventional, profit-maximizing enterprise.
13 See Frydman, Gray, Hessel and Rapaczynski (1999), where the revenue performance of such firms, not of interest on the present occasion, has also been analyzed.
26
Appendix 1. The graphical presentation of the LMF/PMF δ dominance ratio for the three types of wage functions14
30
1
X’
yWn
We
Wz
Z
W
LL LeLz
E
Mym
my
Figure A1.1 The LMF and PMF dominance regions, identified with the Sy and SП subfamilies of the discrete S family, are approximated by the darker and lighter shaded areas, bordered by the Wz, Wn, and We functions: The case of linear labor’s marginal product and linear wage functions, δ12 = 3.64, see table 1 of Appendix 2. X = 2L – 0.2L2 - production function X’ = 2 – 0.4L - labor’s marginal product y = 2 – 0.2L – C/L - income per worker C = 0.68 - fixed costs We = 0.4 - entry wage horizontal line, ak = 0 ak = n a∆ k - the (varying) labor scarcity parameter ak, where ∆ ak = a∆ , k = 1,…,n Wk = We + akL2 - typical wage function Wn = We + anL - neutral wage function that implies equal equilibrium of LMF and PMF, an = 0.160 Wz = We + azL - zero-profit wage function that implies zero-profit and equal equilibrium of LMF and PMF, az = 0.743
14 Note that in Appendix 1 the entry-wage is We = 0.4, while in both Appendix 1 and Appendix 2 the product price is p=1. Also, as already mentioned in part V, the maximum point of income per worker is the same in all figures: M = ( , yy
mL m) ≈ (1.98, 1.26).
27
Figure A1.2 The LMF and PMF dominance regions, identified with the Sy and SП subfamilies
, X’, y , C,We, ak, and Wk - as in figure A1.1 tion, defined as in figure A1.1, an = 0.085
of the discrete S family, are approximated by the darker and lighter shaded areas, bordered by the Wz, Wn, and We functions: The case of linear labor’s marginal product and quadratic wage functions, δ22 = 10.9, see table 1 of Appendix 2. XWn = We + anL2 - neutral wage funcWz = We + azL - zero-profit wage function, defined as in figure A1.1, a2
z = 1.01
28
0 3
1
Figure A1.3 The LMF and PMF dominance regions, identified with the Sy and SП subfamilies
, X’, y , C,We, ak, and Wk - as in figure A1.1 tion, defined as in figure A1.1, an = 0.0347
ppendix 2. The values of the LMF/PMF δ dominance ratio and of the average size
of the discrete S family, are approximated by the darker and lighter shaded areas, bordered by the Wz, Wn, and We functions: The case of linear labor’s marginal product and cubic wage functions, δ32 = 45.0, see table 1 of Appendix 2. XWn = We + anL3 - neutral wage funcWz = We + azL - zero-profit wage function, defined as in figure A1.1, a3
z = 1.61 A
15 of the LMF dominance region )( ySN
. The values of the δ dominance ratio, resulted from 27 numerical simulations
A
15 The values of δ = δij (i,j=1,2,3), and thus of )( ySN , are approximate where, as already mentioned in footnote 11 above, the computational error can be made arbitrarily small.
29
We = 0.4
T1 X’ = 3.5 – 0.6L2
y = 3.5 – 0.2L2 – C/L C = 2.85
T2 X’ = 2 – 0.4L
y = 2 – 0.2L – C/L C = 0.680
T3 X’ = 2.08/(1.45L0.2)
y = (2.6/1.45L0.2) – C/L C = 0.600
an = 0.231 az = 0.505
an = 0.160 az = 0.743
an = 0.227 az = 0.980
S1 Wn = We + an L Wz = We + az L δ11 = 1.19 δ12 = 3.64 δ13 = 3.32
an = 0.0803az = 0.344
an = 0.0850 az = 1.01
an = 0.0790 az = 1.86
S2 Wn = We + an L2
Wz = We + az L2δ21 = 3.28 δ22 = 10.9 δ23 = 22.5
an = 0.0314 az = 0.261
an = 0.0350 az = 1.61
an = 0.0314 az = 4.15
S3 Wn = We + an L3
Wz = We + az L3δ31 = 7.31 δ32 = 45.0 δ33 = 131
Table 1. The values of the δ dominance ratio for the entry-wage We = 0.4 Most of the above dominance ratios δ = δij (i,j=1,2,3), where δ is of Definition 2 of part V, and where the entry-wage is We = 0.4, already appear in part V. Technologies T1, T2 and T3, and the wage functions S1, S2 and S3, are also defined in part V, respectively in sections 1 and 2.
We = 0.63
T1 X’ = 3.5 – 0.6L2
y = 3.5 – 0.2L2 – C/L C = 2.85
T2 X’ = 2 – 0.4L
y = 2 – 0.2L – C/L C = 0.6800
T3 X’ = 2.08/(1.45L0.2)
y = (2.6/1.45L0.2) – C/L C = 0.600
an = 0.171 az = 0.363
an = 0.172 az = 0.490
an = 0.166 az = 0.628
S1 Wn = We + an L Wz = We + az L δ11 = 1.12 δ12 = 1.85 δ13 = 2.78
an = 0.0595 az = 0.231
an = 0.0620 az = 0.564
an = 0.0580 az = 1.00
S2 Wn = We + an L2
Wz = We + az L2δ21 = 2.88 δ22 = 8.10 δ23 = 16.2
an = 0.0232 az = 0.161
an = 0.0250 az = 0.756
an = 0.0230 az = 1.86
S3 Wn = We + an L3
Wz = We + az L3δ31 = 5.94 δ32 = 29.2 δ33 = 79.9
Table 2. The values of the δ dominance ratio for the entry-wage We = 0.63 Most of the above dominance ratios δ = δij (i,j=1,2,3), where δ is of Definition 2 of part V and where the entry-wage is We = 0.63, already appear in part V. Technologies T1, T2 and T3, and the wage functions S1, S2 and S3, are also defined in part V, respectively in sections 1 and 2.
30
We = 1
T1 X’ = 3.5 – 0.6L2
y = 3.5 – 0.2L2 – C/L C = 2.85
T2 X’ = 2 – 0.4L
y = 2 – 0.2L – C/L C = 0.680
T3 X’ = 2.08/(1.45L0.2)
y = (2.6/1.45L0.2) – C/L C = 0.600
an = 0.0750 az = 0.149
an = 0.0720 az = 0.168
an = 0.0690 az = 0.195
S1 Wn = We + an L Wz = We + az L δ11 = 0.987 δ12 = 1.33 δ13 = 1.83
an = 0.0260 az = 0.085
an = 0.0260 az = 0.136
an = 0.0240 az = 0.204
S2 Wn = We + an L2
Wz = We + az L2δ21 = 2.27 δ22 = 4.23 δ23 = 7.50
an = 0.0102 az = 0.0510
an = 0.0110 az = 0.124
an = 0.00900 az = 0.245
S3 Wn = We + an L3
Wz = We + az L3δ31 = 4.00 δ32 = 10.3 δ33 = 26.2
Table 3. The values of the δ dominance ratio for the entry-wage We = 1 Most of the above dominance ratios δ = δij (i,j=1,2,3), where δ is of Definition 2 of part V, and where the entry-wage is We = 1, already appear in part V. The technologies T1, T2 and T3, and the wage functions S1, S2 and S3, are also defined in part V, respectively in sections 1 and 2. B. The δ average dominance ratio, resulted from 27 numerical simulations
(A2.1) ∑ ∑= =
==3
1
3
11.16
271
i jijδδ ,
where δijs are of tables 1-3. C. The average size of the LMF dominance subfamily, )( ySN , obtained via δ of (A2.1)
(A2.2) 942.01
)( =+
=δ
δySN ,
where (A2.2) is analogous to (33c) of the text.
31
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